The Mandelbrot set is built by iterations of function (map)
zm+1 = f( zm ) = zm2 + c or
fc: zo -> z1 -> z2 -> ...
for complex z and c. Iterations begin from starting point zo (usually zo = 0 + 0 i).
|For real c and zo , zm are real too
and we can trace iterations on 2D (x,y) plane. To plot the first
iteration we draw vertical red line from xo = 0 toward blue
curve y = f(x) = x2 + c, where y1 =
f(xo) = c.
drag mouse to change the C value
|To get the second iteration we draw red horizontal line to the green y = x line, where x1 = y1 = f(xo). Then draw again vertical line to the blue curve to get y2 = f(x1) and so on. Dependence of xm on m is plotted in the right part of this applet. Points fc: xo -> x1 -> x2 -> ... at some value c and xo form an orbit of xo.|
|After 25 steps iterations go near to an attracting
fixed point x* = f(x*) of the map
f. f doesn't move the point. Fixed points correspond to
intersections of y = x and y = f(x) (green and blue) curves.
There are always two fixed points for a quadratic map f because of
two roots of quadratic equation
f(x*) - x* = x*2 + c - x* = 0,
x1,2 = 1/2 -+ (1/4 - c)1/2
The second fixed point (the right intersection) is repelling.
|The roots may be complex for some c values.
You can see here that for c > 1/4
attracting fixed point becomes repelling (and even complex
therefore we can not see intersections on real plane). Iterations go
to infinity. It is proven that an orbit go to infinity if |zn|
Such qualitative change in iteration dynamics is called bifurcation.
Points on complex plane c that starting from zo = 0 don't go to infinity under iterations form the Mandelbrot set.
|For c = -0.75 attracting fixed point becomes repelling and iterations converge to attracting period-2 orbit x1 -> x2 -> x1 ... (see Birth of attracting period 2 orbit later).|